CAREER: Computation, Combinatorics, and Reality in Algebraic Geometry, with Applications

Sottile将从事的项目包括代数几何和代数组合学的研究，加强代数几何与应用科学之间的联系，以及教育推广。他在代数几何方面的工作将集中在旗子流形及其舒伯特变种的具体方面，涉及一系列问题，从真正的舒伯特微积分，从量子舒伯特微积分的扩展，从舒伯特微积分中的组合问题，以及从舒伯特变种的几何。这将加深我们对代数几何中计算、组合学和实数现象的理解。Sottile将研究某些组合Hopf代数的结构性质，以及应用于多面体、舒伯特演算和物理中的某些组合方案的工作。他积极从事计算代数几何在其他领域的应用，特别是离散几何和应用科学。这项科学工作的一个统一主题是利用计算机实验来研究具体的数学对象，为理论工作提供信息。虽然代数几何关注的是关于方程解的抽象问题，但它对其他学科的价值主要是通过具体的对象和计算工具。该项目的一项主要活动是通过研究合作、组织会议和为应用科学家共同编写代数几何教科书，帮助将现代计算工具从代数几何传播到其他领域，包括应用科学。Sottile还将为马萨诸塞州西部有才华的高中生组织一个拓展项目，其特别目标是满足附近城市地区学生的需求，其中许多学生来自传统上在科学领域代表性不足的群体。这项工作的目的：研究、技术转让、服务和推广，是为索泰尔人一生的综合研究和教育活动提供一个坚实的基础。

Sottile will work on projects ranging from research in algebraic geometry and algebraic combinatorics, to strengthening links between algebraicgeometry and applied science, to educational outreach. His work in algebraic geometry will focus on concrete aspects of flag manifolds and their Schubertvarieties, involving a range of problems from the real Schubert calculus, from extensions of quantum Schubert calculus, from combinatorial problems in Schubert calculus, and from the geometry of Schubert varieties. This will deepen our understanding of computation, combinatorics, and real-number phenomena in algebraic geometry. Sottile will study structural properties ofcertain combinatorial Hopf algebras, work that has applications to polytopes, to the Schubert calculus, and to certain combinatorial schemes in physics. He is actively engaged in applications of computational algebraic geometry to other fields,particularly discrete geometry and the applied sciences. A unifying theme in this scientific work is studying concrete mathematical objects using computer experimentation to inform theoretical work.While algebraic geometry is concerned with abstract questions about solutions to equations, its value to other disciplines is largely through concrete objects and computational tools. A major activity of this project is to help disseminate the modern computational tools from algebraic geometry to other fields, including the applied sciences, through research collaboration, organizing conferences, and co-authoring a textbook on algebraic geometry for applied scientists. Sottile will also organize an outreach program for talented high school students in Western Massachusetts with the particular goal ofserving the needs of students in nearby urban areas, many of whom comefrom groups traditionally underrepresented in science. The intention of this work: the research, the technology transfer, the service and the outreach, is to provide a firm foundation for a lifetime of integrated research and educational activities for Sottile.